Package 'QuantileGH'

Quantile Least Mahalanobis Distance Estimator for Tukey $g$-&-$h$ Mixture

Description

Tools for simulating and fitting finite mixtures of the 4-parameter Tukey $g$-&-$h$ distributions. Tukey $g$-&-$h$ mixture is highly flexible to model multimodal distributions with variable degree of skewness and kurtosis in the components. The Quantile Least Mahalanobis Distance estimator QLMDe is used for estimating parameters of the finite Tukey $g$-&-$h$ mixtures. QLMDe is an indirect estimator that minimizes the Mahalanobis distance between the sample and model-based quantiles. A backward-forward stepwise model selection algorithm is provided to find

• a parsimonious Tukey $g$-&-$h$ mixture model, conditional on a given number-of-components; and

• the optimal number of components within the user-specified range.

Author(s)

Maintainer: Tingting Zhan [email protected] (ORCID) [copyright holder]

Other contributors:

• Inna Chervoneva [email protected] (ORCID) [contributor, copyright holder]

Examples

# see ?QLMDe

---

Naive Estimates of Finite Mixture Distribution via Clustering

Description

Naive estimates for finite mixture distribution fmx via clustering.

Usage

fmx_cluster(
  x,
  K,
  distname = c("GH", "norm", "sn"),
  constraint = character(),
  ...
)

Arguments

x	numeric vector, observations
K	integer scalar, number of mixture components
distname	character scalar, name of parametric distribution of the mixture components
constraint	character vector, parameters ($g$ and/or $h$ for Tukey $g$-&-$h$ mixture) to be set at 0. See function fmx_constraint for details.
...	additional parameters, currently not in use

Details

First of all, if the specified number of components $K\geq 2$, trimmed $k$-means clustering with re-assignment will be performed; otherwise, all observations will be considered as one single cluster. The standard $k$-means clustering is not used since the heavy tails of Tukey $g$-&-$h$ distribution could be mistakenly classified as individual cluster(s).

In each of the one or more clusters,

• letterValue-based estimates of Tukey $g$-&-$h$ distribution (Hoaglin, 2006) are calculated, for any $K\geq 1$, serving as the starting values for QLMD algorithm. These estimates are provided by function fmx_cluster.

• the median and mad will serve as the starting values for $\mu$ and $\sigma$ (or $A$ and $B$ for Tukey $g$-&-$h$ distribution, with $g = h = 0$), for QLMD algorithm when $K = 1$.

Value

Function fmx_cluster returns an fmx object.

---

Best Naive Estimates for Finite Mixture Distribution

Description

Best estimates for finite mixture distribution fmx.

Usage

fmx_hybrid(x, test = c("logLik", "CvM", "KS"), ...)

Arguments

x	numeric vector, observations
test	character scalar, criteria for selecting the optimal estimates. See Details.
...	additional parameters of functions fmx_normix and fmx_cluster

Details

Function fmx_hybrid compares Tukey $g$-&-$h$ mixture estimate provided by function fmx_cluster and the normal mixture estimate by function fmx_normix, and select the one either with maximum likelihood (test = 'logLik', default), with minimum Cramer-von Mises distance (test = 'CvM') or with minimum Kolmogorov distance (Kolmogorov_fmx).

Value

Function fmx_hybrid returns an fmx object.

Examples

library(fmx)
d1 = fmx('norm', mean = c(1, 2), sd = .5, w = c(.4, .6))
set.seed(100); hist(x1 <- rfmx(n = 1e3L, dist = d1))
fmx_normix(x1, distname = 'norm', K = 2L)
fmx_normix(x1, distname = 'GH', K = 2L)

(d2 = fmx('GH', A = c(1,6), B = 2, g = c(0,.3), h = c(.2,0), w = c(1,2)))
set.seed(100); hist(x2 <- rfmx(n = 1e3L, dist = d2))
fmx_cluster(x2, K = 2L)
fmx_cluster(x2, K = 2L, constraint = c('g1', 'h2'))
fmx_normix(x2, K = 2L, distname = 'GH')
fmx_hybrid(x2, distname = 'GH', K = 2L)

---

Naive Parameter Estimates using Mixture of Normal

Description

Naive parameter estimates for finite mixture distribution fmx using mixture of normal distributions.

Usage

fmx_normix(x, K, distname = c("norm", "GH", "sn"), alpha = 0.05, R = 10L, ...)

Arguments

x	numeric vector, observations
K	integer scalar, number of mixture components
distname	character scalar, name of parametric distribution of the mixture components
alpha	numeric scalar, proportion of observations to be trimmed in trimmed $k$-means algorithm tkmeans
R	integer scalar, number of normalmixEM replicates
...	additional parameters, currently not in use

Details

fmx_normix ... the cluster centers are provided as the starting values of $\mu$'s for the univariate normal mixture by EM algorithm. R replicates of normal mixture estimates are obtained, and the one with maximum likelihood will be selected

Value

Function fmx_normix returns an fmx object.

---

Quantile Least Mahalanobis Distance estimates

Description

The quantile least Mahalanobis distance algorithm estimates the parameters of single-component or finite mixture distributions by minimizing the Mahalanobis distance between the vectors of sample and theoretical quantiles. See QLMDp for the default selection of probabilities at which the sample and theoretical quantiles are compared.

The default initial values are estimated based on trimmed $k$-means clustering with re-assignment.

Usage

QLMDe(
  x,
  distname = c("GH", "norm", "sn"),
  K,
  data.name = deparse1(substitute(x)),
  constraint = character(),
  probs = QLMDp(x = x),
  init = c("logLik", "letterValue", "normix"),
  tol = .Machine$double.eps^0.25,
  maxiter = 1000,
  ...
)

Arguments

x	numeric vector, the one-dimensional observations.
distname	character scalar, name of mixture distribution to be fitted. Currently supports 'norm' and 'GH'.
K	integer scalar, number of components (e.g., must use 2L instead of 2).
data.name	character scalar, name for the observations for user-friendly print out.
constraint	character vector, parameters ($g$ and/or $h$ for Tukey $g$-&-$h$ mixture) to be set at 0. See function fmx_constraint for details.
probs	numeric vector, percentiles at where the sample and theoretical quantiles are to be matched. See function QLMDp for details.
init	character scalar for the method of initial values selection, or an fmx object of the initial values. See function fmx_hybrid for more details.
tol, maxiter	see function vuniroot2
...	additional parameters of optim

Details

Quantile Least Mahalanobis Distance estimator fits a single-component or finite mixture distribution by minimizing the Mahalanobis distance between the theoretical and observed quantiles, using the empirical quantile variance-covariance matrix quantile_vcov.

Value

Function QLMDe returns an fmx object.

See Also

fmx_hybrid

Examples

data(bmi, package = 'mixsmsn')
hist(x <- bmi[[1L]])
QLMDe(x, distname = 'GH', K = 2L)

---

Forward Selection of the Number of Components $K$

Description

To compare $gh$-parsimonious models of Tukey $g$-&-$h$ mixtures with different number of components $K$ (up to a user-specified $K_\text{max}$) and select the optimal number of components.

Usage

QLMDe_stepK(
  x,
  distname = c("GH", "norm"),
  data.name = deparse1(substitute(x)),
  Kmax = 3L,
  test = c("BIC", "AIC"),
  direction = c("forward", "backward"),
  ...
)

Arguments

x	numeric vector, observations
distname, data.name	character scalars, see parameters of the same names in function QLMDe
Kmax	integer scalar $K_\text{max}$, maximum number of components to be considered. Default 3L
test	character scalar, criterion to be used, either Akaike's information criterion AIC, or Bayesian information criterion BIC (default).
direction	character scalar, direct of selection in function step_fmx, either 'forward' (default) or 'backward'
...	additional parameters

Details

Function QLMDe_stepK compares the $gh$-parsimonious models with different number of components $K$, and selects the optimal number of components using BIC (default) or AIC.

The forward selection starts with finding the $gh$-parsimonious model (via function step_fmx) at $K = 1$. Let the current number of component be $K^c$. We compare the $gh$-parsimonious models of $K^c+1$ and $K^c$ component, respectively, using BIC or AIC. If $K^c$ is preferred, then the forward selection is stopped, and $K^c$ is considered the optimal number of components. If $K^c+1$ is preferred, then the forward selection is stopped if $K^c+1=K_{max}$, otherwise update $K^c$ with $K_c+1$ and repeat the previous steps.

Value

Function QLMDe_stepK returns an object of S3 class 'stepK', which is a list of selected models (in reversed order) with attribute(s) 'direction' and 'test'.

Examples

data(bmi, package = 'mixsmsn')
hist(x <- bmi[[1L]])
QLMDe_stepK(x, distname = 'GH', Kmax = 2L)

---

Percentages for Quantile Least Mahalanobis Distance estimation

Description

A vector of probabilities to be used in Quantile Least Mahalanobis Distance estimation (QLMDe).

Usage

QLMDp(
  from = 0.05,
  to = 0.95,
  length.out = 15L,
  equidistant = c("prob", "quantile"),
  extra = c(0.005, 0.01, 0.02, 0.03, 0.97, 0.98, 0.99, 0.995),
  x
)

Arguments

from, to	numeric scalar, minimum and maximum of the equidistant (in probability or quantile) probabilities. Default .05 and .95, respectively
length.out	non-negative integer scalar, the number of the equidistant (in probability or quantile) probabilities.
equidistant	character scalar. If 'prob' (default), then the probabilities are equidistant. If 'quantile', then the quantiles (of the observations x) corresponding to the probabilities are equidistant.
extra	numeric vector of additional probabilities, default c(.005, .01, .02, .03, .97, .98, .99, .995).
x	numeric vector of observations, only used when equidistant = 'quantile'.

Details

The default arguments of function QLMDp returns the probabilities of c(.005, .01, .02, .03, seq.int(.05, .95, length.out = 15L), .97, .98, .99, .995).

Value

A numeric vector of probabilities to be supplied to parameter p of Quantile Least Mahalanobis Distance QLMDe estimation). In practice, the length of this probability vector p must be equal or larger than the number of parameters in the distribution model to be estimated.

Examples

library(fmx)
(d2 = fmx('GH', A = c(1,6), B = 2, g = c(0,.3), h = c(.2,0), w = c(1,2)))
set.seed(100); hist(x2 <- rfmx(n = 1e3L, dist = d2))

# equidistant in probabilities
(p1 = QLMDp()) 

# equidistant in quantiles
(p2 = QLMDp(equidistant = 'quantile', x = x2))

---

Re-Assign Observations Trimmed Prior to Trimmed $k$-Means Clustering

Description

Re-assign the observations, which are trimmed in the trimmed $k$-means algorithm, back to the closest cluster as determined by the smallest Mahalanobis distance.

Usage

reAssign(x, ...)

## S3 method for class 'tkmeans'
reAssign(x, ...)

Arguments

x	a tkmeans object
...	potential parameters, currently not in use.

Details

Given the tkmeans input, the mahalanobis distance is computed between each trimmed observation and each cluster. Each trimmed observation is assigned to the closest cluster (i.e., with the smallest Mahalanobis distance).

Value

Function reAssign.tkmeans returns an 'reAssign_tkmeans' object, which inherits from tkmeans class.

Note

Either kmeans or tkmeans is slow for big x.

Examples

library(tclust)
data(geyser2)
clus = tkmeans(geyser2, k = 3L, alpha = .03)
plot(clus, main = 'Before Re-Assigning')
plot(reAssign(clus), main = 'After Re-Assigning')

---

Forward Selection of $gh$-parsimonious Model with Fixed Number of Components $K$

Description

To select the $gh$-parsimonious mixture model, i.e., with some $g$ and/or $h$ parameters equal to zero, conditionally on a fixed number of components $K$.

Usage

step_fmx(
  object,
  test = c("BIC", "AIC"),
  direction = c("forward", "backward"),
  ...
)

Arguments

object	fmx object
test	character scalar, criterion to be used, either Akaike's information criterion AIC-like, or Bayesian information criterion BIC-like (default).
direction	character scalar, 'forward' (default) or 'backward'
...	additional parameters, currently not in use

Details

The algorithm starts with quantile least Mahalanobis distance estimates of either the full mixture of Tukey $g$-&-$h$ distributions model, or a constrained model (i.e., some $g$ and/or $h$ parameters equal to zero according to the user input). Next, each of the non-zero $g$ and/or $h$ parameters is tested using the likelihood ratio test. If all tested $g$ and/or $h$ parameters are significantly different from zero at the level 0.05 the algorithm is stopped and the initial model is considered $gh$-parsimonious. Otherwise, the $g$ or $h$ parameter with the largest p-value is constrained to zero for the next iteration of the algorithm.

The algorithm iterates until only significantly-different-from-zero $g$ and $h$ parameters are retained, which corresponds to $gh$-parsimonious Tukey $g$-&-$h$ mixture model.

Value

Function step_fmx returns an object of S3 class 'step_fmx', which is a list of selected models (in reversed order) with attribute(s) 'direction' and 'test'.

See Also

step

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